Graduation Semester and Year

2014

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Minerva Cordero

Abstract

It is well known that any finite semifield, S, can be viewed as an n-dimensional vector space over a finite field or prime order, Fp, and that the multiplication in S defines and can be defined by an n x n x n cubical array of scalars, A. For any element a E S, the matrix, La, corresponding to left multiplication by a can be determined from A. In this paper we show that there exists a unique monic polynomial of minimal degree, f E Fp[x], such that f(a) = 0, and which divides the minimal polynomial of La. Furthermore, we show that some properties of f in Fp[x] correspond to properties of a in S. These results, in turn, help optimize a method we introduce which uses A to determine the automorphism group of S. We show that under certain conditions A can be inflated to define a new semifield, S[m], over the field Fpm , and that inflation preserves isotopism and isomorphism between inflated semifields. Finally, we apply our results to the 16-element semifields, and give algebraic constructions for each of these semifields for which no construction currently exists.

Disciplines

Mathematics | Physical Sciences and Mathematics

Comments

Degree granted by The University of Texas at Arlington

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Mathematics Commons

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